$\operatorname{curl} u =\nabla \times u$
$=\left|\begin{array}{ccc}
e _1 & e _2 & e _3 \\
\frac{\partial}{\partial x_1} & \frac{\partial}{\partial x_2} & \frac{\partial}{\partial x_3} \\
u_1 & u_2 & u_3
\end{array}\right|$
Answer (1)
The curl of a vector field u is denoted as curl u = ∇ × u .
It is calculated as the determinant of a matrix with unit vectors e 1 , e 2 , e 3 , partial derivatives ∂ x 1 ∂ , ∂ x 2 ∂ , ∂ x 3 ∂ , and vector field components u 1 , u 2 , u 3 .
The determinant is expanded to obtain the components of the curl along each axis.
The final expression for the curl is: curl u = ( ∂ x 2 ∂ u 3 − ∂ x 3 ∂ u 2 ) e 1 + ( ∂ x 3 ∂ u 1 − ∂ x 1 ∂ u 3 ) e 2 + ( ∂ x 1 ∂ u 2 − ∂ x 2 ∂ u 1 ) e 3 .
Explanation
Introduction to Curl We are given the formula for the curl of a vector field u , denoted as curl u or ∇ × u . The objective is to explain this formula.
Definition of Curl The curl of a vector field u is defined as the determinant of a matrix. This matrix is constructed using unit vectors, partial derivatives, and the components of the vector field.
First Row: Unit Vectors The first row of the matrix consists of the unit vectors e 1 , e 2 , and e 3 , which represent the directions along the x 1 , x 2 , and x 3 axes, respectively.
Second Row: Partial Derivatives The second row contains the partial differential operators ∂ x 1 ∂ , ∂ x 2 ∂ , and ∂ x 3 ∂ . These operators represent the rate of change with respect to the coordinates x 1 , x 2 , and x 3 .
Third Row: Vector Field Components The third row consists of the components of the vector field u , denoted as u 1 , u 2 , and u 3 . These components are functions of x 1 , x 2 , and x 3 .
Calculating the Determinant The determinant of the matrix is calculated as follows:
curl u = ( ∂ x 2 ∂ u 3 − ∂ x 3 ∂ u 2 ) e 1 + ( ∂ x 3 ∂ u 1 − ∂ x 1 ∂ u 3 ) e 2 + ( ∂ x 1 ∂ u 2 − ∂ x 2 ∂ u 1 ) e 3
This expression gives the curl of the vector field u in terms of its components and their partial derivatives.
Summary of Curl In summary, the curl of a vector field measures the infinitesimal rotation of the vector field at a point. It is a vector quantity, with its direction indicating the axis of rotation and its magnitude indicating the strength of the rotation.
Examples
The curl is used in fluid dynamics to describe the vorticity of a fluid. For example, the curl of the velocity field of a hurricane indicates the strength and direction of the rotation of the storm. In electromagnetism, the curl of the magnetic field is related to the electric current density, as described by Ampere's Law. Understanding the curl helps engineers design efficient turbines and understand electromagnetic phenomena.
